Usually, you install OpenERP and as part of the installation, you have all of the 'core' modules on your system.

You don't need to install 'core' modules via 'Apps', you can remove the filter under the 'Installed Modules' view and install from there.

Apps shows all modules available on apps.openerp.com, which includes the 'core' ones installed with the software (this is to support updates).

When you install a module from Apps that isn't already on your server, it gets downloaded to /server/openerp/addons

Apps is a service that is part of the Enterprise (code identical) version of OpenERP and is optional.

There are a TON of existing questions here that explain more about the Apps service.

When you install a new module you've new functionality on your system. For example if you are installing sale module, all the sale dependencies are going to be installed. This creates new tables on the database and enable the module logic, workflows, functionality, accounting.... Well there is so much functionality, but you've to install it to do that.